Gravitation

GRAVITATION SESSION – 01 AIM  Introduction  Universal Law of Gravitation  Important characteristics of gravitational force 1]

INTRODUCTION The motion of celestial bodies such as the sun, the moon, the earth and the planets etc. has been a subject of fascination since time immemorial. Indian astronomers of the ancient times have done brilliant work in this field, the most notable among them being Arya Bhatt the first person to assert that all planets including the earth revolve round the sun. A millennium later the Danish astronomer Tyco Brahe (15461601) conducted a detailed study of planetary motion which was interpreted by his pupil Johan’s Kepler (1571-1630), ironically after the master himself had passed away. Kepler formulated his important findings in three laws of planetary motion

2]

UNIVERSAL LAW OF GRAVITATION According to this law "Each particle attracts every other particle. The force of attraction between them is directly proportional to the product of their masses and inversely proportional to square of the distance between them". F∝

or F =

where G = 6.67 × 10–11 Nm2 kg–2 is the universal gravitational constant. This law holds good irrespective of the nature of two objects (size, shape, mass etc.) at all places and all times. That is why it is known as universal law of gravitation. Dimensional formula of g : =

=

[

]

= [M–1 L3 T–2 ]

Newton law of gravitation in vector form: ⃗=





& ⃗

=





Where ⃗ is the force on mass m1 exerted by mass m2 and vice-versa. = − ̂ , Thus ⃗ we get ⃗ = − ⃗ Now ̂

=

̂ . Comparing above,

3]

IMPORTANT FORCE

CHARACTERISTICS

OF

GRAVITATIONAL

i]

Gravitational force between two bodies form an action and reaction pair i.e. the forces are equal in magnitude but opposite in direction.

ii]

Gravitational force is a central force i.e. it acts along the line joining the centers of the two interacting bodies.

iii] Gravitational force between two bodies is independent of the nature of the medium, in which they lie. iv] Gravitational force between two bodies does not depend upon the presence of other bodies. v]

Gravitational force is negligible in case of light bodies but becomes appreciable in case of massive bodies like stars and planets.

vi] Gravitational force is long range - force i.e., gravitational force between two bodies is effective even if their separation is very large. For example, gravitational force between the sun and the earth is of the order of 1027 N although distance between them is 1.5 × 107 km

CLASS EXERCISE: 1] The centers of two identical spheres are at a distance 1.0 m apart. If the gravitational force between them is 1.0 N, then find the mass of each sphere. (G = 6.67 × 10–11 m3 kg–1 sec–1) . Sol. Gravitational force F = on substituting F = 1.0 N , r = 1.0 m and G = 6.67 × 10–11 m3 kg–1 sec–1 we get m = 1.225 × 105 kg 2]

Two particles of masses m1 and m2, initially at rest at infinite distance from each other, move under the action of mutual gravitational pull. Show that at any instant their relative )/ , where R is their velocity of approach is 2 ( + separation at that instant. Sol. The gravitational force of attraction on m1 due to m2 at a separation r is F1 = Therefore, the acceleration of m1 is a1 =

=

Similarly, the acceleration of m2 due to m1 is a2 = – the negative sign being put as a2 is directed opposite to a1. The relative acceleration of approach is a = a1– a2 = If

(



)

is the relative velocity, then a =

.... (1) =



.

But – = v (negative sign shows that r decreases with increaing t ).

∴ a = –

v.

.... (2)

From (1) and (2), we have v dv = –

(

)

dr

Integrating, we get At r = ∞, v = 0 (given), and so C = 0. ( ) ∴ v2 = Let v = vR when r = R. Then vR =

=

(

(

)

= +C

)

3]

Three identical bodies of mass M are located at the vertices of an equilateral triangle with side L. At what speed must they move if they all revolve under the influence of one another's gravity in a circular orbit circumscribing the triangle while still preserving the equilateral triangle ? Sol. Let A, B and C be the three masses and O the centre of the circumscribing circle. The radius of this circle is R = sec 30° = ×

=

√

√

.

Let v be the speed of each mass M along the circle. Let us consider the motion of the mass at A. The force of gravitational attraction on it due to the masses at B and C are along AB and along AC The resultant force is therefore 2



30° =

√

along AD

This, for preserving the triangle, must be equal to the necessary centripetal force. That is,

√

=

=

√

[Θ R = L/√3] or

=

4] 5]

Find out the time period of circular motion in above example A solid sphere of lead has mass M and radius R.A spherical hollow is dug out from it (see figure). Its boundary passing through the centre and also touching the boundary of the solid sphere. Deduce the gravitational force on a mass m placed at P, which is distant r from O along the line of centres. Sol. Let O be the centre of the sphere and O' that of the hollow (figure). For an external point the sphere behaves as if its entire mass is concentrated at its centre. Therefore, the gravitational force on a mass `m` at P due to the original sphere (of mass M) is

= , along PO. The diameter of the smaller sphere (which would be cut off) is R, so that its radius OO' is R/2. The force on m at P due to this sphere of mass M' (say) would be F¢ = G

along PO¢.

[Θ distance PO¢ = r – ]

As the radius of this sphere is half of that of the original sphere, we have

M¢ = . ∴ F¢ = G



along PO¢.

As both F and F¢ point along the same direction, the force due to the hollowed sphere is F – F¢ =

–

=

1−

.

SESSION - 02 AIM  Gravitational Field  Gravitational potential  Relation Between Gravitational Field and Potential 1]

GRAVITATIONAL FIELD The space surrounding the body within which its gravitational force of attraction is experienced by other bodies is called gravitational field. Gravitational field is very similar to electric field in electrostatics where charge 'q' is replaced by mass 'm' and electric constant 'K' is replaced by gravitational constant 'G'. The intensity of gravitational field at a point is defined as the force experienced by a unit mass placed at that point. E=

=

The unit of the intensity of gravitational field is N kg–1. In vector form ⃗ = − ̂

Dimensional formula of intensity of gravitational field = =

[ ]

=[

]

2]

GRAVITATIONAL POTENTIAL The gravitational potential at a point in the gravitational field of a body is defined as the amount of work done by an external agent in bringing a body of unit mass from infinity to that point, slowly (no change in kinetic energy). Gravitational potential is very similar to electric potential in electrostatics. Let the unit mass be displaced through a distance dr towards mass M, then work done is given by dW = F dr

=

. ⇒ ∫

gravitational potential,

=−

= ∫



=

. Thus

.

The unit of gravitational potential is J kg –1. Dimensional Formula of gravitational potential =

3]

=

[ ]

= [M°L2 T–2].

RELATION BETWEEN POTENTIAL

GRAVITATIONAL

FIELD

AND

The work done by an external agent to move unit mass from a point to another point in the direction of the field E, slowly through an infinitesimal distance dr = Force by external agent × distance moved = – Edr. Thus

= –



⇒ E=–

.

Therefore, gravitational field at any point is equal to the negative gradient at that point. CLASS EXERCISE 1]

Find the relation between the gravitational field on the surface of two planets A & B of masses mA, mB & radius RA & RB respectively if (i) they have equal mass (ii) they have equal (uniform) density Let EA & EB be the gravitational field intensities on the surface of planets A & B. then, EA =

=

Similarly, = EB =





=

=



RB

(i) for mA = mB

=

(ii) For & A = B

=



The gravitational field in a region is given by ⃗ = – (20N/kg) ( ̂ + ̂). Find the gravitational potential at the origin (0, 0) – (in J/kg) a) zero b) 20√2 c) – 20√2 d) cannot be defined Sol: V = – ∫ . = [∫ . + ∫ . ] = 20x + 20y at origin V = 0 2]

3]

In above problem, find the gravitational potential at a point whose co-ordinates are (5, 4) – (in J/kg) a) – 180 b) 180 c) – 90 d) zero Sol: V = 20 × 5 + 20 × 4 = 180 J/kg 4]

In the above problem, find the work done in shifting a particle of mass 1 kg from origin (0, 0) to a point (5, 4) – (In J) a) – 180 b) 180 c) – 90 d) zero Sol: W = m (V¦ – Vi) = 1 (180 – 0) = 180J 5]

There are two bodies of masses 100 kg and 10000 kg separated by a distance 1m. At what distance from the smaller body, the intensity of gravitational field will be zero a) b) c) d)

6]

Which one of the following graphs represents correctly the variation of the gravitational field (F) with the distance (r) from the centre of a spherical shell of mass M and radius a I

a)

r =a

r

b)

I

I

I

r =a

r

c)

r =a

r

d)

r =a

r

7]

The curve depicting the dependence of intensity of gravitational field on the distance r from the centre of the earth I

I

O

I

r=R O

r

8]

I

O

O

r=R

r

r=R

r

r=R

r

a) b) c) d) A thin spherical shell of mass M and radius R has a small hole. A particle of mass m is released at the mouth of the hole. Then m

R

M

a) The particle will execute simple harmonic motion inside the shell b) The particle will oscillate inside the small, but the oscillations are not simple harmonic c) The particle will not oscillate, but the speed of the particle will go on increasing d) None of these

SESSION – 03 AIM Gravitational Potential & Field for Different Objects  Ring  Linear mass  Infinite Linear mass  Uniform solid sphere  uniform thin spherical shell  uniform thick spherical shell GRAVITATIONAL POTENTIAL AND FIELD FOR DIFFERENT OBJECTS I]

Ring =

&

(

)

=(

) /

̂ or

= –

Gravitational field is maximum at a distance, is –2GM/3 − √





= ± /√2 and it

II] A linear mass of finite length on its axis

a]

Potential ⟹

b]

=−

ln(

+

)=−

ln

√

Field intensity ⟹

=−

=

√

III] An infinite uniform linear mass distribution of linear mass density , Here

=

And noting that

=

in case of a finite rod

we get, for field intensity

=

Potential for a mass-distribution extending to infinity is not defined. However even for such mass distributions potentialdifference is defined. Here potential difference between points P1 and P2 respectively at distances d1 and d2 from the infinite rod, = 2 ln

IV] Uniform solid sphere a]

Point P inside the shell ≤ , then

b]

=−

(3

=–

and

−

) &

=–

=0

Point P outside the shell ≥ , then

=−

&

= –

V] Uniform thin spherical shell a]

Point P Inside the shell ≤ , then

=

&

=0

, and at the centre

b]

Point P outside shell ≥ , then

=

&

= −

VI] Uniform thick spherical shell a] Point outside the shell =–

;

=–

b] Point inside the Shell =–

=0 c] Point between the two surface = –

;

= –

CLASS EXERCISE 1] Calculate the gravitational field intensity at the center of the base of a hollow hemisphere of mass M and radius R. (Assume the base of hemisphere to be open) Sol: We consider the shaded elemental ring of mass, ( ) =( 2 ) Field due to this ring at 0, = (see formulae for field due to a ring)

or, Hence, 2]

= =∫

=∫



or,

=

Calculate the gravitational field intensity and potential at the centre of the base of a solid hemisphere of mass m, radius R. Sol: We consider the shaded elemental disc of radius R and thickness ( ) ( ) Its mass, =

Or = sin Field due to this plate at O, (

=

)

(

) (



Or

=

=

−

(see field due to a uniform disc) )

+

∴ or

=∫

=∫

(

)

=

Now potential due to the element under consideration at the centre of the base of the hemisphere, = – (see potential due to a circular plate) or, ∴

(

=

(

∫ (

=–

= –

−

) )

)

−

+

or, =– Aliter : Consider a hemispherical shell of radius r and thickness dr

Its mass,

=

(2

) or,

=

Since all points of this hemispherical shell are at the same distance r from O. Hence potential at O due to it is, = = ∴ =∫ =

SESSION – 04 AIM  Gravitational potential Energy  Gravitational self-energy 1]

GRAVITATIONAL POTENTIAL ENERGY Gravitational potential energy of two mass system is equal to the work done by an external agent in assembling them, while their initial separation was infinity. Consider a body of mass placed at a distance from another body of mass M. The gravitational force of attraction between them is given by, = . Now, Let the body of mass m is displaced from point. C to B through a distance 'dr' towards the mass M, then work done by internal conservative force (gravitational) is given by,

=

=

⟹∫

=∫

∴ Gravitational potential energy,

=−

Special cases i]

From above equation, it is clear that gravitational potential energy of two mass system increases with increase in separation (r) (i.e. it becomes less negative).

ii]

Gravitational P.E. becomes maximum (or zero) at

= ∞.

iii] If the body of mass m moves from a distance r1 to r2 (r1 > r2), then work done or change in gravitational P.E. is given by =∫

=

=−

∫

−

Since r1 > r2, so change in gravitational P.E. of the body is negative. It means, when the body is brought near to the earth, P.E. of the earth-mass system decrease. iv] When the body of mass m is moved from the surface of earth (i.e., r1 = Re) to a height ℎ (i.e., r2 = Re + h), then change in P.E. of the earth-mass system s given by = –

−

=

=

1−

1− 1+

Using binomial expansion ⟹ Since

= =

1− 1− then

=

≈ ℎ

Gravitational potential difference is defined as the work done by an external agent to move a unit mass from one point to the other point in the gravitational field. According to the definition, E is the force experienced by a unit mass at A. The direction of this force is towards the body of mass M. Now the work done to move the unit mass from A to B is given by = ⃗. ⃗ =



180° = –

This work done is equal to the gravitational potential difference (dV). Where 2]

is called potential gradient.

GRAVITATIONAL SELF-ENERGY The gravitational self-energy of a body (or a system of particles) is defined as the work done by an external agent in assembling the body (or system of particles) from infinitesimal elements (or particles) that are initially an infinite distance apart.

Gravitational self energy of a system of n particles Potential energy of n particles at an average distance 'r' due to their mutual gravitational attraction is equal to the sum of the potential energy of all pairs of particle, i.e., ∑

=–



This expression can be written as

=−

∑

∑

If consider a system of ′ ′ particles, each of same mass ′ ′ and seperated from each other by the same average distance ′ ′, then self energy Or

=−

∑

∑

Thus on the right hand side ′ ′ comes ′ ′ times while ′ ′ comes – 1 times. Thus =−

( − 1)

Gravitational self energy of a uniform sphere (star)

=−

(

= – (4 )2 4

)

where r = ,

Ustar= – (4 )2 ∫ = – (4 )2 ∴

= –

= –



.



CLASS EXERCISE: 1]

Calculate the velocity with which a body must be thrown vertically upward from the surface of the earth so that it may reach a height of 10 R, where R is the radius of the earth and is equal to 6.4 × 108 m. (Earth's mass = 6 × 1024 kg, Gravitational constant G = 6.7 × 10–11 nt-m2/kg2) Sol. The gravitational potential energy of a body of mass m on earth's surface is ( ) = − where M is the mass of the earth (supposed to be concentrated at its centre) and R is the radius of the earth (distance of the particle from the centre of the earth). The gravitational energy of the same body at a height 10 R from earth's surface, i.e. at a distance 11R from earth's centre is (11 ) = – change in potential energy (11 )– ( ) = – −

−

=

This difference must come from the initial kinetic energy given to the body in sending it to that height. Now, supose the body is

thrown up with a vertical speed v, so that its initial kinetic 2 = energy is 2. Then = . Putting the given values : =

×

×( ×

. × ( . ×



)

)



= 1.07 × 104 / . 2] Distance between centres of two stars is 10 a. The masses of these stars are M and 16 M and their radii are a & 2a resp. A body is fired straight from the surface of the larger star towards the smaller star. What should be its minimum initial speed to reach the surface of the smaller star? Sol. Let P be the point on the line joining the centres of the two planets s.t. the net field at it is zero

Then,

−(

)

= 0 ⇒ (10 a– r)2 = 16 r2

⇒ 10 − = 4 ⇒ = 2 Potential at point P, . = −( = − = )

..

Now if the particle projected from the larger planet has enough energy to cross this point, it will reach the smaller planet.

For this, the K.E. imparted to the body must be just enough to raise its total mechanical energy to a value which is equal to P.E. at point P. i.e.

−

or, or, 3]

4]

5]

6]

(

)

=

=− 2 =



,

=

Escape velocity of a body of 1 kg mass on a planet is 100 m/sec. Gravitational potential energy of the body at the planet is a) – 5000 J b) – 1000 J c)– 2400 J d) 5000 J A body of mass m rises to a height ℎ = from the earth’s surface where R is earth’s radius. If g is acceleration due to gravity at the earth’s surface, the increase in potential energy is a) mgh b) ℎ c) ℎ d) ℎ The work done is bringing three particles each of mass 10 g from large distances to the vertices of an equilateral triangle of side 10 cm. a)1 × 10 b)2 × 10 c) 4 × 10 d)1 × 10 The potential energy due to gravitational field of earth will be maximum at a) Infinite distance b) The poles of earth c) The centre of earth d) The equator of earth

SESSION - 05 AIM:   

Acceleration due to gravity Variation of Acceleration due to Gravity Escape speed

1]

ACCELERATION DUE TO GRAVITY It is the acceleration, a freely falling body near the earth’s surface acquires due to the earth’s gravitational pull. The property by virtue of which a body experiences or exerts a gravitational pull on another body is called gravitational mass mG, and the property by virtue of which a body opposes any change in its state of rest or uniform motion is called its inertial mass mI thus if → is the gravitational field intensity due to the earth at a point P, and is acceleration due to gravity at the same point, then →= →. Now the value of inertial & gravitational mass happen to be exactly same to a great degree of accuracy for all bodies. Hence, → = → The gravitational field intensity on the surface of earth is therefore numerically equal to the acceleration due to gravity

(g), there. Thus we get,

= where , Me = Mass of earth Re = Radius of earth Note Here the distribution of mass in the earth is taken to be spherical symmetrical so that its entire mass can be assumed to be concentrated at its center for the purpose of calculation of g. 2]

VARIATION OF ACCELERATION DUE TO GRAVITY a]

Effect of altitude Acceleration due to gravity on the surface of the earth is given by, = Now, consider the body at a height 'h' above the surface of the earth, then the acceleration due to gravity at height 'h' given by

ℎ = (

)

=

1+



1−

The decrease in the value of 'g' with height ℎ = – ℎ =

.

when h gequator. The weight of

the body increase as the body taken from the equator to the pole.

d] Effect of rotation of the earth The earth rotates around its axis with angular velocity . Consider a particle of mass m at latitude . The angular velocity of the particle is also .

According to parallelogram law of vector addition, the resultant force acting on mass m along PQ is F=[( {2

) +(

=

cos } cos(180 − )]

×

= [(

cos ) +

) +(

cos ) − (2

1+

cos

−2



cos ) cos ]

cos

At pole = 90° ⇒ gpole = g , At equator

= 0 ⇒



= 1−

.

Hence gpole > gequator If the body is taken from pole to the equator, then ¢ =

1−

.

Hence % change in weight =

× 100 =

× 100

(

)

× 100 =

3]

ESCAPE SPEED The minimum speed required to project a body from the surface of the earth so that it never returns to the surface of the earth is called escape speed. A body thrown with escape speed goes out of the gravitational pull of the earth. Work done to displace the body from the surface of the earth (r = Re) to infinity ( = ∞ ) is given by ∫

= ∫

or

= –

= ⇒ =

∫



= –





−



Let ve be the escape speed of the body of mass m, then kinetic energy of the body is given by =

⇒ ve =

2

= 11.2



.

Important points 1] Escape speed depends on the mass and size of the planet. That is why escape velocity on the Jupiter is more than on the earth. 2] Escape speed is independent of the mass of the body. 3] Any body thrown upward with escape speed start moving around the sun. CLASS EXERCISE : 1]

If R is the radius of the earth and g the acceleration due to gravity on the earth’s surface, the mean density of the earth is a) b) c) d)

2]

A mass ‘m’ is taken to a planet whose mass is equal to half that of earth and radius is four times that of earth. The mass of the body on this planet will be a) b) c) d)

3]

The diameters of two planets are in the ratio 4 : 1 and their mean densities in the ratio 1: 2. The acceleration due to gravity on the planets will be in ratio a) 1 : 2 b) 2 : 3 c)2 : 1 d) 4 : 1 The acceleration due to gravity on the moon is only one sixth that of earth. If the earth and moon are assumed to have the same density, the ratio of the radii of moon and earth will be a) b) c) d)

4]

( )

5]

( )

A body weight W Newton at the surface of the earth. Its weight at a height equal to half the radius of the earth will be

a) 6]

7]

8]

b)

c)

d)

The value of g on the earth’s surface is 980 cm/sec2. Its value at a height of 64 km from the earth’s surface is (Radius of the earth R = 6400 Kilometers) a) 960.40 / 2 b) 984.90 / 2 c) 982.45 / 2 d) 977.55 / 2 The decrease in the value of g at height h from earth’s surface is a) b) c) d) If the value of ‘g’ acceleration due to gravity, at earth surface is 10 / 2 , its value in / 2 at the centre of the earth, which is assumed to be a sphere of radius ‘R’ metre and uniform mass density is a) 5 b) c) d) Zero

SESSION - 06 & 07 AIM:  Motion of Satellites and Kepler Laws  Satellite Speed(or Orbital speed)  Geo-stationary Satellites or Geo-synchronous Satellites  Launching of an Artificial Satellite around The earth 1]

MOTION OF SATELLITES AND KEPLER LAWS A heavenly body revolving around a planet in an orbit is called natural satellite. For example, moon revolves around the planet the earth, so moon is the satellite of the earth. Their motions can be studied with the help of kepler's laws, as stated: i]

Law of orbit Each Planet moves around the sun in an elliptical orbit with the sun at one of the foci as shown in figure. The eccentricity of an ellipse is defined as the ratio of the distance SO and AO i.e. e =

=

=

The distance of closest approach with the sun at F1 is AS. This distance is called perigee. The greatest distance (BS) of the planet from the sun is called apogee. Perigee (AS) = AO – OS = a – ea = a (1 – e) Apogee (BS) = OB + OS = a + ea = a (1 + e)

ii] Law of Areas The line joining the sun and a planet sweeps out equal areas in equal intervals of time. A planet takes the same time to travel from A to B as from C to D as shown in figure. (The shaded areas are equal). Naturally the planet has to move faster from C to D. The law of areas is identical with the law of conservation of angular momentum. Areal velocity = Hence

2

=

(

)

= 2

=

= constant.

iii] Law of periods The square of the time for the planet to complete a revolution about the sun is proportional to the cube of semi major axis of the elliptical orbit.

i.e. Centripetal force = Gravitational force =

⇒

=

Now, speed of the planet is =











Substituting value in above equation

=

⇒

=



or

Since

is constant,

∴ 2 ∝ 3 2]

=



= constant

SATELLITE SPEED The speed required to put the satellite into its orbit around the earth is called orbital speed. The gravitational attraction between satellite and the earth provides the necessary centripetal force.

(

)

=(

)

or

=(

)



,

=

(

)

=

(

. When h